Genus 2 curve 249939.a.249939.1

• Label: 249939.a.249939.1
• Conductor: $249939$
• Discriminant: $-249939$
• Mordell-Weil group: \(\Z/{2}\Z\)
• Sato-Tate group: $\mathrm{USp}(4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 + x^2y = x^5 + x^4 - 45x^3 - 10x^2 + 541x - 357$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(249939\)	\(=\)	\( 3^{3} \cdot 9257 \)
Discriminant:	\( \Delta \)	\(=\)	\(-249939\)	\(=\)	\( - 3^{3} \cdot 9257 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(135960\)	\(=\)	\( 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 103 \)
\( I_4 \)	\(=\)	\(222901920\)	\(=\)	\( 2^{5} \cdot 3^{2} \cdot 5 \cdot 19 \cdot 8147 \)
\( I_6 \)	\(=\)	\(11622651537195\)	\(=\)	\( 3^{2} \cdot 5 \cdot 258281145271 \)
\( I_{10} \)	\(=\)	\(-999756\)	\(=\)	\( - 2^{2} \cdot 3^{3} \cdot 9257 \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_2$

Rational points

All points: \((1 : 0 : 0),\, (-21 : -882 : 4)\)

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z\)

Generator	$D_0$						Height	Order
\((-21 : -882 : 4) - (1 : 0 : 0)\)	\(4x + 21z\)	\(=\)	\(0,\)	\(32y\)	\(=\)	\(-441z^3\)	\(0\)	\(2\)

2-torsion field: 4.2.3999024.3

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(3\)
Regulator:	\( 1 \)
Real period:	\( 1.118838 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 4.475354 \)
Analytic order of Ш:	\( 16 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(3\)	\(3\)	\(1\)	\(1 - T\)
\(9257\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 102 T + 9257 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.30.3	yes

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$\mathrm{USp}(4)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).